Symmetric quadrature rules for simplexes based on sphere close packed lattice arrangements

•A general approach for computing optimal symmetric quadrature rules on simplexes is identified.•A new family of optimal symmetric quadrature rules on the 2-simplex (triangle) is identified.•A new optimal symmetric quadrature rule on the 3-simplex (tetrahedron) is identified.

Sphere close packed (SCP) lattice arrangements of points are well-suited for formulating symmetric quadrature rules on simplexes, as they are symmetric under affine transformations of the simplex unto itself in 2D and 3D. As a result, SCP lattice arrangements have been utilized to formulate symmetric quadrature rules with Np=1Np=1, 4, 10, 20, 35, and 56 points on the 3-simplex (Shunn and Ham, 2012). In what follows, the work on the 3-simplex is extended, and SCP lattices are employed to identify symmetric quadrature rules with Np=1Np=1, 3, 6, 10, 15, 21, 28, 36, 45, 55, and 66 points on the 2-simplex and Np=84Np=84 points on the 3-simplex. These rules are found to be capable of exactly integrating polynomials of up to degree 1717 in 2D and up to degree 99 in 3D.